Electrical Impedance Tomography (EIT) has been studied by a large number of authors and in most of this work it has been considered to be a two-dimensional ...
An Image Reconstruction Algorithm for Three Dimensional Electrical Impedance Tomography A. Le Hyaric and M. K. Pidcock School of Computing & Mathematical Sciences, Oxford Brookes University, Headington, Oxford, OX3 0BP, UK.
Abstract Electrical Impedance Tomography (EIT) has been studied by a large number of authors and in most of this work it has been considered to be a two-dimensional problem. Many groups are now turning their attention to the full three-dimensional case in which the computational demands become much greater. It is interesting to look for ways to reduce this demand and in this paper we describe an implementation of an algorithm that is able to achieve this by pre-computing many of the quantities needed in the image reconstruction. The algorithm is based on a method called NOSER introduced some years ago by Cheney et al [3] but we have extended their method by introducing a more realistic electrode model into the analysis.
1. Introduction In practical applications the inverse conductivity problem is known as Electrical Impedance Tomography (EIT). EIT has been extensively studied by a large number of authors (see for example [1]) and substantial progress has been made both mathematically and practically. Much of this work has considered the EIT image reconstruction problem to be essentially two-dimensional. This assumption has been recognised as almost certainly incorrect but it has enabled the simplification of the technology used in EIT and reduced the computational demands, thus enabling researchers to concentrate on other issues. Now, however, many groups are turning their attention to the full three-dimensional problem. In this case the computational demands become much greater and it is interesting to look for ways in which this demand can be reduced. Consequently, we have investigated one algorithm that is able to achieve this by transferring much of the computational effort to a pre-reconstruction time and thereby make three dimensional image reconstruction reasonably rapid. This algorithm is based on a method introduced some years ago by the EIT group at Rensselaer Polytechnic Institute [2], [3] and [4]. It is referred to as Newton’s One-Step Error Reconstructor or NOSER because it reconstructs the conductivity distribution within the object by taking just one step of Newton’s method for solving non-linear equations [5]. In NOSER, the conductivity is described in terms of N constants and the objective of the image reconstruction algorithm is to determine these constants. NOSER uses a constant conductivity distribution (corresponding to all N constants being equal) as an initial guess and consequently, many of the calculations involved in the reconstruction are analytical and the conductivity update is obtained by inverting one N × N matrix system. In our implementation of NOSER, the current drive electrodes are represented using the “gap” model [6]. This takes into account the discretization effects of the electrodes that are neglected by the “continuum” model [6]. We combine this model with trigonometric current patterns and find the potential in a body of constant conductivity in terms of a Fourier series. In this paper, we describe the calculations necessary for our implementation of NOSER and present some results using simulated data.
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2. The method Let Ω be a right cylinder of radius R , height H and boundary ∂Ω . If u is the electric potential and σ the electric conductivity, then the governing boundary value problem is given by
1 0 5 0 56 = 0 ∂u0 x 5 = j0 x 5 σ 0 x5 ∇. σ x ∇u x
∂n
x ∈Ω
(1a)
x ∈ ∂Ω ,
(1b)
where n denotes the unit outward normal to ∂Ω and j denotes the applied current density. For convenience, we will work with the resistivity ρ = 1 σ . In order to satisfy the conservation of charge and to select a unique solution, we respectively require that
I
∂Ω
05
j x =
I
∂Ω
05
u x = 0.
On the curved boundary of the cylinder we attach a number of electrodes {Γl } of equal size symmetrically
spaced and placed in a number of horizontal bands. We use ∆θ and ∆h to denote the angular width and height of an electrode, Lv to represent the number of bands and Lh the number of electrodes per band. In this paper, the subscript v stands for “vertical” and the subscript h for “horizontal”. Given the current input and voltage measurements on these electrodes, our goal is to reconstruct the resistivity distribution within the cylinder. To ensure a good spatial resolution in the image, we apply a set of L − 1 different current patterns Gk I = I1k , I 2k ,!, I Lk , k = 1,!, L − 1 through the L = Lv Lh current drive electrodes and make corresponding
7 0G
2
2
5
7
voltage measurements V = V , V , !, VLk on L voltage electrodes. We make the voltage measurements at the centre of the current electrodes and we apply a basis of trigonometric current patterns. This defines
I lk
k
k 1
k 2
Î È kvS zl Ø Ñcos É Ù Ñ Ê H Ú ÑÑ È kvS zl Ø Ïcos É Ù cos khT l Ñ Ê H Ú Ñ Èk Sz Ø ÈÈ L Ø Ø Ñcos É v l Ù sin É É kh h Ù T l Ù ÊÊ 2Ú Ú ÑÐ Ê H Ú
1
kv
1,! , Lv 1 ; kh
kv
0,1,! , Lv 1 ; kh
1, 2,! , Lh 2
kv
0,1,! , Lv 1 ; kh
Lh 2 1,! , Lh 1
0 (2)
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with R, θ l , z l the cylindrical co-ordinates of the centre of the l th current electrode relative to the origin taken to be at the centre of the base of the cylinder. The particular arrangement of electrodes considered in this paper means that θ l =
We
describe
(
) where l
2π l * − 1 Lh the
resistivity
*
in
= l (mod Lh ) . terms
of
a
finite
number
05 2 05 05
of
parameters
;ρ @ and
write
0 57 the voltage pattern that is
G ρ = ( ρ1 , ρ 2 ," , ρ N ) ∈ \ N . We denote by U k ρ = U1k ρ , U 2k ρ ,!, U Lk ρ
n
produced on ∂Ω by the k current pattern for a particular resistivity estimate U . Our inverse problem thus becomes that of finding the optimal resistivity ρ that minimises some measure of the differences th
05
G G V k − U k ρ for k = 1,2, !, L − 1.
One way of achieving this is to use the method of least squares [5] which consists of minimising the error 2
05
05
L −1 G G E ρ = ∑ Vk −Uk ρ k =1
2
L −1 L
0 57 .
2
= ∑ ∑ Vl k − U lk ρ k =1 l =1
2
We obtain a system of N non-linear equations in N variables by noting that, at a minimum
05
0 = Fn ρ =
05
0 57 0 5
2
L −1 L ∂E ρ ∂U lk ρ = −2∑ ∑ Vl k − U lk ρ , ∂ρ n ∂ρ n k =1 l =1
n = 1,2, !, N .
(3)
Given a resistivity estimate U 0 , Newton’s method solves this system by computing a new estimate
G 1 G U 0 ËÍ F U 0 ÛÝ F U 0 ,
U1 G
(4)
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where F ′ ρ is the N × N Jacobian matrix L −1 L ∂U k ρ ∂U k ρ L −1 L ∂ 2U lk ( ρ ) ∂ ∂E ( ρ ) l ( ) l ( ) k k ′ (ρ ) = . = 2∑∑ − 2∑∑ Vl − U l ( ρ ) Fnm ∂ρ m ∂ρ n ∂ρ m ∂ρ n ∂ρ m ∂ρ n k =1 l =1 k =1 l =1
(
)
(5)
G
We iterate this process until either the values of U 1 and U 0 are close enough or the value of F U 0
is
sufficiently near zero. Newton’s algorithm is widely used because it is more rapidly convergent than most of the other root-finding methods. However, it requires two function evaluations for each iteration step which in our case are very computationally expensive. In order to minimise algorithm execution time NOSER takes only one step of Newton’s method. It starts from a constant resistivity estimate ρ 0 and produces the first Newton estimate ρ 1 . This approach was first introduced by Simske [2]. In [3], Cheney et al give a full description of the algorithm in two dimensions and in [4] Goble et al extend it to three dimensions. As we start from ρ 0 = constant, computationally expensive terms in
G G F ρ 0 and F ′ ρ 0 are dependent only on the geometry and can be pre-computed and stored. Consequently, the main computation becomes the relatively inexpensive solution of an N × N linear system.
2 7
2 7
′ ( ρ ) we replace the term We use a Levenberg-Marquardt approximation [5] and in Fnm L −1 L
∂ 2U lk ( ρ )
k =1 l =1
∂ρ m ∂ρ n
β nm = − 2∑∑ (Vl k − U lk ( ρ ))
where δ is the Krönecker delta and α nm = 2
L −1 L
∑∑ k =1 l =1
by β nm = γα nmδ nm ,
∂U lk ( ρ ) ∂U lk ( ρ ) ∂ρ m
∂ρ n
,
for n, m = 1, 2,! , N .
′ ( ρ ) becomes Thus, the regularised version of Fnm ′ ( ρ ) = α nm + γα nmδ nm . Fnm
(6)
The regularisation parameter γ is chosen empirically to generate suitable stability and image contrast.
3
L −1 L
∑∑ (V U (1)) k
We write the initial resistivity in the form ρ 0 = ρ * (1,1," ,1) ∈ \ N , where ρ * =
k =1 l =1 L −1 L
k l
l
∑∑ (U (1)) k =1 l =1
k l
2
∈ , since
it is well known [3] that this is the constant resistivity distribution predicting voltage measurements that best fit the measured data.
3. The gap model In [3] and [4], NOSER is used in conjunction with the “continuum ” electrode model. This model for boundary condition (1b) assumes that the current density on the boundary is a continuous function and this greatly simplifies the implementation of NOSER. Here, we make use of the more realistic “gap” model that includes the effect of discretization of the current patterns by assuming that the current density is constant over the surface of each electrode and zero in all the gaps between electrodes. With this assumption boundary condition (1b) becomes
j (x) = σ ( x)
∂u ( x ) ∂n
Il ∆θ∆H = 0
x ∈ Γl
l = 1,! , L (7)
L
x ∉ * Γl , l =1
where we denote the l th electrode by Γl .
05
In order to implement the NOSER algorithm defined by equations (4) and (6) we need to determine U lk 1 , that th
is the voltage observed on the l voltage electrode when the resistivity in the cylinder is unity. To achieve this, we start by expanding the current density j k defined by equations (2) and (7) into a Fourier series
j k (θ , z ) =
2
∞ ∞ mπ ∂u k ( R,θ , z ) = ∑∑ Anmk cos ( nθ ) + Bnmk sin ( nθ ) cos ∂r H n =0 m =0
(
)
z,
7
k k , Bnm are real constants to be determined. where Anm
For a cylinder of constant resistivity ρ * , problem (1) can be solved analytically using the method of separation of variables to give
u k r ,T , z
È mS r Ø In É Ê H ÙÚ H U È mS z Ø k k Anm cos nT Bnm sin nT cos É ÇÇ Ê H ÙÚ 'T'H n 0 m 1 mS È mS R Ø In É Ê H ÙÚ *
U* 'T'H
rn
Ç nR A n 1
k n0
cos nT Bnk0 sin nT ,
n 1
( )
The voltage at the centre of the l th electrode (needed in equation (4)) is U lk ρ 0 = u k ( R,θ l , zl ) .
4
(8)
< A
< A
k k 4. Calculation of the Fourier coefficients Anm and Bnm
< A < A corresponding to
k k and Bnm In this section we will give explicit expressions for the Fourier coefficients Anm
the current density j k defined by equations (2) and (7). It is straightforward to show that they can be written in kh
kh
kv
kv
k k the form Anm = X n Z m and Bnm = Yn Z m , where Lh
l 1
and
Ç Pl kh anl ibnl , Z mkv
X nkh iYnkh
Lv
ÇQ
m ≥ 1, n ≥ 0
kv l l m
c ,
l 1
'H , H
a0l ib0l
'T , 2S
anl ibnl
2 È n'T Ø inTl sin É e , Ê 2 ÙÚ nS
cml
(
Qlkv
c0l
)
Pl kh = trig kh*θ l ,
(9a)
4 È mS zl Ø È mS'H Ø , m, n ≥ 1 sin É cos É Ù Ê 2H Ú Ê H ÙÚ mS Èk Sz Ø cos É v l Ù , Ê H Ú
(9b)
L kh* = kh mod h 2
.
where trig = sin or cos (depending on the form of the current pattern defined by the values of k h and k v ). The symmetry of our geometry allows us to further simplify these expressions which we will illustrate by k
k
calculting X n h . To account for both cosine and sine current patterns we now denote pl h
2 7
e
ik *hT l
so that for
Lh −1
Im plkh , and for trig = cos, Pl kh = Re plkh . Denote also xnkh = ∑ plkh anl .
trig = sin, Pl kh
l =0
l 0
By replacing a by its expression given by equation (9a), we obtain :
∆θ x = 2π kh 0
Lh
∑e l =1
ik *hθ l
∆θ = 2π
Lh
∑e
i
2π k *h l Lh
.
(10)
l =1
Similarly, 2π k l L 2π nl i Lh h 2 n∆θ h sin x = e ∑ cos nπ L 2 l =1 h *
kh n
2π nl 2π nl 2π kh l L −i i L 1 n∆θ h i Lh Lh e h = + sin e e ∑ 2 nπ l =1 *
2π l * 2π l * L Lh i (k −n ) 1 n∆θ h i Lh (k h + n ) Lh h sin = + e e ∑ ∑ nπ 2 l =1 l =1
It is straightforward to see that if O is an integer then
5
(11)
Lh 1 i 2SO l Lh
O =2p,
Î Lh Ï Ð0
Çe l 0
p
0,1, 2,"
otherwise
.
Using this result in equations (10) and (11), we find that
x 0kh = 0 xnkh =
k h*
since
(
)
Lh n∆θ 1 + δ k * 0 sin h nπ 2
,
x nkh = 0
Lh 2
n pLh kh*
0,1, 2,!
p
otherwise.
2 7
2 7 for
The above expressions provide X n h , n ≥ 0 by writing X n h = Im x n h for trig = sin and X n h = Re x n h k
k
k
k
trig = cos. Similar results can be derived for Ynkh , Z mkv for m, n ≥ 0 .
k
The infinite sums discussed above can only be approximated by truncating the summations at finite values of the summation index. If we denote by N max and M max the truncation indices in the Fourier series (8) we see that a
1
6
number Lh N max + 1 of values of X n h , n ≥ 0 and of Yn h , n ≥ 0 need to be calculated. In fact, the expressions k
k
we derived for X n h , n ≥ 0 and Yn h , n ≥ 0 show that at most N max + 1 values are non-zero. In the same way, k
k
we have found that at most M max + 1 values of Z mv , m ≥ 0 are different from zero. Consequently, most of k
< A < A
k k and Bnm are zero which means that calculations involving the Fourier series these Fourier coefficients Anm (8) are achievable at reasonable computational cost.
5. The calculation of
2 7
∂U k ρ 0 ∂ρ n
Equations (3), (4) and (6) show that we need to compute the matrix
2 7.
∂U k ρ 0 ∂ρ n
expanding the matrix in terms of the trigonometric basis given in equation (2).
U k I , U n O
U k U n
L 1
Ç
O 1
IO,IO
I O , where we denote M , N
L
ÇM N l
l 1
By using (2), we find that
6
l
.
We can achieve this by
Î Lv 2 È svS zl Ø Ñ Lh Ç cos ÉÊ Ù, H Ú Ñ l1 Lv ÑÑ Lh 2 2 È svS zl Ø , Ï Ç cos shT l Ç cos É Ê H ÙÚ l 1 Ñ l1 Ñ Lh Lh Ø Ø Lv ÈsSz Ø 2 ÈÈ Ñ Ç sin É É sh Ù T l Ù Ç cos 2 É v l Ù , Ê H Ú ÊÊ 2Ú Úl1 ÑÐ l 1
Is, Is
Lh
∑ cos2 ( shθ l ) and
We compute the sums
l =1
Lh
∑ sin
2
l =1
Lh sh − 2
sv
1,! , Lv 1 ; sh
sv
0,1,! , Lv 1 ; sh
1, 2,! , Lh 2
sv
0,1,! , Lv 1 ; sh
Lh 2 1,! , Lh 1
0
θ l using the results [7]
n cos n 1 x sin nx 2 2sin x
n
Ç cos kx 2
k 1
n cos n 1 x sin nx 2 2sin x
n
Ç sin kx 2
k 1
This gives :
Is, Is =
Lv π zl Lh 2 + + δ δ 1 sh 0 Lh ∑ cos sv sh 2 H 2 l =1
,
where again δ is the Krönecker delta. In the usual implementation of the NOSER algorithm we model the resistivity distribution by dividing the cylinder into N voxels in which we assume the resistivity to be constant. In this case it has been shown in [3] that
Is,
∂U k ∂ρ n
≈
I
∇u k .∇u s ,
Mn
where M n is the n th mesh element and u k is given by equation (8).
6. Numerical results In this section, we present two images reconstructed by our 3D algorithm using numerical data. We consider a cylinder of radius 15 cm and 26 cm high. On the boundary of the cylinder, we place 4 bands of 16 electrodes that are equally spaced in each of the bands and 5 cm square. The cylinder is discretized into 464 voxels and the background resistivity is set to 9 Ω − m. In the first configuration, a target of resistivity 0 +4 Ω − m is introduced into the cylinder. This target is 6.3 cm high, 3 cm long in the radial direction and covers π 4 radians in the angular direction going from angle −π 8 to angle π 8 radians. The regularization parameter γ is set to 5.
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Figure 1 shows the distribution of resistivity within the cylinder between − π 8 and 9π 8 . The mesh that is shown in this figure is not the element mesh used in NOSER but is included from the graphics routine to aid clarity. The resistivity distribution reconstructed by our 3D software is shown in Figure 2 between the same angles. Figure 2 ranges from 8 to 14.125 Ω − m. The image clearly shows the presence of the target and a good contrast in resistivity between the background and the target especially considering its small size.
Figure 1 : Resistivity distribution to be reconstructed
Figure 2 : Resistivity distribution reconstructed by our 3D algorithm
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In the second configuration, two targets are introduced into the cylinder. The resistivity in the target placed on the right is 0 +4 Ω − m . This target is 13 cm high, 5 cm long in the radial direction and covers π 4 radians in the angular direction going from angle −π 8 to angle π 8 radians. The resistivity in the second target is 0 −4 Ω − m. It is 6.5 cm high, 5 cm long in the radial direction and covers π 4 radians in the angular direction going from angle 7π 8 to angle 9 π 8 radians. The regularization parameter γ is set to 1. Figure 3 shows the distribution of resistivity within the cylinder between − π 8 and 9π 8 radians. The resistivity distribution reconstructed by our 3D software is shown in Figure 4 between the same angles. The resistivity values in Figure 4 range from –15.58 to 33.02 Ω − m. We can clearly see the presence of the targets in the reconstructed image. To illustrate the stability of our algorithm, we present in Figure 5 the resistivity distribution it has reconstructed when a 10% random noise has been added to the voltage measurement data.. In Figure 5 ranges the resistivity values vary from –15.58 to 33.02 Ω − m.
Figure 3 : Resistivity distribution to be reconstructed
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Figure 4 : Resistivity distribution reconstructed by our 3D algorithm
Figure 5 : Resistivity distribution reconstructed by our 3D algorithm using randomly noisy data
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7. Conclusions In this work we have extended previous work on the three dimensional NOSER algorithm to include the ‘gap’ electrode model. We have seen how the use of symmetry can reduce much of the work needed in the precalculations. The images obtained using this method are quite reasonable and the method can be extended to include the ‘hybrid’ electrode arrangement where the voltage measurements are made in between the current electrodes. After completion of this work we became aware of the similar work undertaken at RPI by R.S. Blue et al [8]. Our results confirm the quality of images obtained using this type of algorithm.
References [1]
Physiol. Meas. 17, Vol 4A, 1996.
[2]
S. J. Simske, “An adaptative current determination and a one-step reconstruction technique for a current tomography system”, M. S. thesis, Rensselaer Polytechnic Institute, Troy, NY, 1987.
[3]
M. Cheney, D. Isaacson, J. C. Newell, S. Simske, and J. C. Goble, “NOSER: An algorithm for solving the inverse conductivity problem”, International Journal of Imaging Systems and Technology, 2:66-75, 1990.
[4]
M. Cheney, D. Isaacson, and J. C. Goble, “Electrical Impedance Tomography in three dimensions”, Applied Computational Electromagnetics Soc. Journal, 7, 128-147 (1992).
[5]
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, “Numerical Recipes : The art of scientific computing”, Cambridge University Press, New York, 1986.
[6]
K.-S. Cheng, D. Isaacson, J. C. Newell, and D. G. Gisser, “Electrode models for electric current computed tomography”, IEEE Trans. Biomed. Eng., 36, 918-924 (1989).
[7]
I.S. Gradshteyn, and I.M. Ryzhik, “Table of integrals, series and products”, Academic Press, Fifth edition, 1994.
[8]
R. S. Blue, D. Isaacson, and J. C. Newell, “Real-time Three-dimensional Electrical Impedance Imaging”, Rensselaer Polytechnic Institute preprint, 1999.
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